Mallat algorithm

An efficient way to implement the concept of QMFs was developed in 1989 by Mal-lat leading to the FWT, which requires only n operations for an -dimensional vector [56]. The Mallat algorithm is in fact a classical scheme known in the signal processing community as a two-channel subband coder. 

Mallat Algorithm is in fact a classical scheme known in the signal processing community as a two-channel subband coder. 

For some bases the calculation of Uy = (4>ilj) matrix will be easy. As we shall see below, the discrete wavelet transform from the Mallat algorithm produces an orthonormal basis which makes U equal to the identity matrix. For orthonormal bases no modification of the original multivariate algorithms is necessary and we can use the method directly on the basis of coefficients C. The conceptual relationship between function, sampled data and the coefficient space is shown in Fig. 1. 

Pillet R, Virieux J (2007) The effects of seismic rotations on inertial sensors. Geophys J Int 171 1314—1323 Rupakhety R et al (2010) Estimating coseismic deformations from Near Source Strong Motion Records Methods and Case Studies. Bull Earthq Eng 8 787-811 Shensa M (1992) The discrete wavelet transform wedding the a trous and mallat algorithms. IEEE Trans Signal Process 40 2464-2482... 

Fig. 40.43. Waveforms for the discrete wavelet transform using the Haar wavelet for an 8-points long signal with the scheme of Mallat s pyramid algorithm for calculating the wavelet transform coefficients.

This is the principle of the pyramidal algorithm developed by Mallat [20], which is computationally more efficient. Continuing the calculations according to this algorithm, the four a components are input to a 4x4 level transformation matrix, giving the level-3 components ... 

There is a whole family of different wavelet methods available depending on the signal properties and the type of information that is to be extracted. However, this chapter will only focus on the fast wavelet transform (FWT) which is based on Mallat s algorithm [39.40]. It should be mentioned that the described methods to achieve parsimonious models are not dependent on one particular type of wavelet transform. Other types of wavelet transforms can be used. FWT is not always optimal for all types of problems and other techniques such as wavelet packets [41], continuous transforms [42,43] and biorthogonal transforms [37] should be considered. Some of the properties of the FWT that makes it an attractive transform are ... 

Fig. 9.13. Mallat s pyramidal algorithm used to implement the DWT. Approximation and detail vectors are indicated by cA and cD respectively subindices denote decomposition level.

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